isprs archives XLII 1 W1 345 2017
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
SIMULATION OF INERTIAL NAVIGATION SYSTEM ERRORS AT AERIAL
PHOTOGRAPHY FROM UAV
R. Shults
Kyiv National University of Construction and Architecture, Faculty for Geoinformation Systems and Territory Management,
Povitroflotskyi Avenue, 31 Kyiv, 03037, Ukraine, shultsrv@gmail.com
Commission V, WG V/7
KEY WORDS: Unmanned aerial vehicle, Aerial photography, Inertial navigation system (INS), Accelerometer systematic shift,
Gyroscope systematic offset, Position accuracy
ABSTRACT:
The problem of accuracy determination of the UAV position using INS at aerial photography can be resolved in two different ways:
modelling of measurement errors or in-field calibration for INS. The paper presents the results of INS errors research by
mathematical modelling. In paper were considered the following steps: developing of INS computer model; carrying out INS
simulation; using reference data without errors, estimation of errors and their influence on maps creation accuracy by UAV data. It
must be remembered that the values of orientation angles and the coordinates of the projection centre may change abruptly due to
the influence of the atmosphere (different air density, wind, etc.). Therefore, the mathematical model of the INS was constructed
taking into account the use of different models of wind gusts. For simulation were used typical characteristics of micro
electromechanical (MEMS) INS and parameters of standard atmosphere. According to the simulation established domination of INS
systematic errors that accumulate during the execution of photographing and require compensation mechanism, especially for
orientation angles. MEMS INS have a high level of noise at the system input. Thanks to the developed model, we are able to
investigate separately the impact of noise in the absence of systematic errors. According to the research was found that on the
interval of observations in 5 seconds the impact of random and systematic component is almost the same. The developed model of
INS errors studies was implemented in Matlab software environment and without problems can be improved and enhanced with new
blocks.
1. INTRODUCTION
The last 25 years in the world, we have seen a stable increase in
requirements for geospatial data. These requirements differ but
generally this is desire to improve the accuracy and detail of
data collection and at the same time increasing the speed and
reducing cost. In the traditional version with the aim of
mapping and GIS projects data were collected using traditional
methods of terrestrial survey or aerial photography. Terrestrial
technologies are complex and energy-intensive and not very
suitable for fast and detailed data collection and updating. On
the other hand, the traditional aerial photography with long
distance from the camera to the object does not allow
completely display all its characteristics, and the results are
highly dependent on weather conditions. Both technologies are
expensive and therefore not very suitable for frequent data
updates. Today, very popular is method of data collection using
GNSS (global navigation satellite system). However, this
method is actually a continuation of traditional terrestrial
survey because it requires direct determination of each point,
although however slightly reducing the cost of work and time
spent on data collection.
An alternative to existing methods of data collection is the use
of technology, which in complex use different navigation
technologies and remote sensing (Bosak, 2014). These
technologies include aerial photography using unmanned aerial
vehicles (UAV). The main advantage of this technology is the
simultaneous reduction of cost and time for data collection. In
comparison with aerial photography, UAV camera equipment is
much simpler, the surveying distances are smaller and the
efficiency is much higher (Colomina et al. 2014). Of course,
aerial photography from the UAVs has its drawbacks, such as
low accuracy at high altitudes. However, in many projects UAV
benefits are more significant in comparison with disadvantages.
Aerial photography from the UAVs provides the accuracy that
requires the most of the topographical work. However,
achieving the required level of accuracy is challenging. The
main problems in obtaining accurate data are concern to
navigation facilities as UAV navigation unit is a complex
system with many sensors that have completely different nature
of information. UAV navigation equipment may include GNSS,
barometric sensor, magnetic compass, tilt sensor, gyroscopes
system, accelerometers systems, INS. The best solution for
aerial photography using UAVs is navigation unit GNSS/INS.
Due to the high cost and bulkiness of accurate inertial
navigation systems for UAV are used miniature
electromechanical navigation systems (MEMS). These INSs
have undeniable advantages from the point of cost and
dimensions. However, the accuracy of inertial systems wishes
to better. Accelerometers and gyroscopes that are part of the
miniature navigation systems have low measurement accuracy.
Since determination of the position and orientation in space,
using INS based on double integration of the measured
accelerations and angular velocities, even minor errors at the
beginning of integration growing very quickly over time. For
INS errors correction are using two approaches:
- The study of a particular model of INS and construction
of mathematical dependencies, which describe system errors.
- INS correction at specified intervals, using GNSS data.
The first approach is suitable for high precision, tactical-grade
INS for which errors are stable and do not change over time.
INS of this type cannot be used for UAV due to high cost and
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
345
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
huge weight. For low-cost INS rational is use of GNSS for
periodic correction (Abdel-Hamid, 2005; Grejner-Brzezinska, et
al. 2004) rather than calibration (Artese et al. 2008), due to
large and unstable errors. The choice of the interval by which
INS correction is performed depends on how quickly
accumulating errors and changing accuracy of INS in intervals
between corrections. Finally, from this depends the accuracy
with which the position of the UAV and the position and
orientation of aerial equipment will be determined (Rehak et al.,
2014). So, important is the task of INS errors simulation and
based on such research, setting INS correction interval using
GNSS. Such researches will allow choosing INS with proper
characteristics. Therefore, the main idea of this paper is
practical research of low-cost INS accuracy by mathematical
simulation results. Based on these results we tried to establish
the influence of INS position on accuracy of topographic maps
creation.
First of all we have to establish the mathematical model of INS.
c cθ
R bn s cθ
-sθ
vE
NH
vN
n
Ωen
M H
vE tg B
N H
where
ie
(1)
= Earth rotation velocity
ab = measured accelerations vector in b-system
ω bib = measured angular velocities vector in b-system
2Ω
n
ie
Ω
n
en
v
n
= vector corrections for influence of
Сoriolis and centrifugal accelerations
γ n = vector of acceleration of normal gravity
0
Ωiee 0 ; Ωien R en Ωiee
e
1
0
M H
1
D 0
N
H c B
0
0
cB
0 ,
ie s B
,
n
enz
n
enx
(3)
n
en y
where B, L, H = geodetic coordinates
vE, vN = east and north components of velocity
M, N = main radius of ellipsoid
Total rotation from e-system to n-system is describing by next
matrix:
s B c L
Ω s L
c B c L
s Bs L
cL
c Bs L
n
en
cB
0
s B
(4)
In expressions, superscript indicates the coordinate system in
which parameters are presented, and the bottom shows the
original system of coordinates. For angular velocities, lower
index indicates in respect of which two coordinate systems
occurs rotation.
Computational model of INS, which is implemented in the nsystem shown in Figure 1. Calculations in this scheme do with
equations (1).
Inertial navigation system
Angular velocities Linear accelerations
D1 v n
rn
n n b
n
n
n
n ,
v
R
a
Ω
Ω
v
γ
2
b
ie
en
n
Rb
n
b
b
R
ω
Ω
b
ib
in
(2)
Rotation from n-system to e-system can be implemented
through the transition velocity:
2. INS MODEL
The main coordinate systems, which are using in inertial
navigation, are inertial system (i-system), earth centered system
(e-system), local horizontal navigation system (n-system), body
coordinate system (b-system). The core description and
determination of these systems you can find in (Biezad , 1999;
Salytcheva, 2004).
The orientation of UAV can be determined by three angles ( φ - heading), which need for connection
roll, θ - pitch,
between vectors in b-system and n-system at the same point in
space.
We chose the INS model in which the angles and coordinates
get in n-system. In such case, we introduce a short description
of this model according to (Zang, 2003).
The equations of position and orientation calculation in nsystem have the next form:
-s cφ+c sθsφ s sφ+c sθcφ
,
c cφ+s sθsφ -c sφ+s sθcφ
cθsφ
cθcφ
a n + vn
аb
R en +
ω bib ω bnb
+
-
ω
R
-
γn
р а ь vа n
ра ітація
n
Ω en
n n
2Ωien v n Ωen
v
e
n
rn
n
Ren , Ωen
+
b
in
R bn
Ωinn
+
Ωien
R en
Ωiee
Figure 1. INS mechanisation (Zang, 2005)
Now we defined INS mathematical model. The next step is to
choose correct model for describing INS errors.
3. INS ERRORS MODEL
ie
The main measuring devices of INS are gyroscopes and
accelerometers. These devices have special errors, which
depend on structure. That is why for follow simulation is
important to choose correct INS errors model. For such model,
we have to choose measurement model for both gyroscopes and
accelerometers.
The equation of measured accelerations in matrix form will
(Park, 2004):
(5)
ab a a s a a N a a γ γ ε a ,
0
.
0
1
Rotation from b-system to n-system is describing by next
matrix:
where a = vector of clear accelerations
a = vector of accelerometers systematic bias
s a = matrix of scale coefficients
Na = matrix of angular accelerometers misalignments
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
346
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
γ = vector acceleration of normal gravity
γ = vector of acceleration of normal gravity variations
ε a = vector of accelerations measurements noise
objects) and complicated curve, which has a form of “eight”
(often is using for INS calibration). All these trajectories
presented below.
Also, equation (5) in expanded form will:
a bX a X a X s X
b
aY aY aY
aZb aZ aZ θ
γ sin θ γ X
γ cos θ sin φ γ
Y
γ cos θ cos φ γ Z
aX
aY
aZ
sY
φ
θ a X
φ aY
sZ aZ
.
The equation of measured angular velocities in matrix form
will:
(6)
ωbib ω ω s ω N ω ε ,
Figure 2. Projection of spatial line on planes XOY and XOZ
where ω = vector of clear angular velocities
ω = vector of gyroscopes systematic bias
s = matrix of scale coefficients
N = matrix of angular gyroscopes misalignments
ε = vector of gyroscopes measurements noise
Also, equation (6) in expanded form will:
b
X
b
Y
b
Z
Y
Z
X
sX
Y
θ
Z
X
sY
φ
θ
φ
sZ
Y
Z
X
Y .
Z
X
In the most cases, the random part is presented as a sum of
white noise and additional components.
(7)
W c r q t,
where
W
c
r
q
t
= white noise
= correlation noise
= random shift
= quantization noise
Figure 3. Closed route along linear object
= trembling noise
The total errors model of random and systematic errors of
accelerometers and gyroscopes for simulation are possible to
present by equations (5) and (6) using devices specifications.
4. INITIAL DATA AND MATLAB SIMULINK MODEL
FOR SIMULATION
For our simulation, we need two types of initial data: technical
specifications of INS and simulated clear accelerations and
angular velocities. In order to construct errors model we chose
follow errors values (Goodall et al. 2012; Barret, 2014), which
presented in table below.
Error
Accelerometer bias a
Axis non-orthogonality
Gyroscope bias
Accelerometers noise, RMS
Gyroscopes noise, RMS
Table 1. INS errors
Value
100 μg, 250 μg, 500 μg
60"
1,5°/hr, 2,5°/hr 5°/hr
5 mg/ hr/√Hг
0.5°/hr/√Hг
According to large value of biases, the scale coefficients of
accelerometers and gyroscopes were neglected.
In the case of measured accelerations and angular velocities
were chosen three types of trajectories: straight spatial line,
classical surveying trajectory for one route (is using for linear
Figure 4. Complicated curve
The movement of UAV has taken uniform, with a constant
speed of 80 km / h. Using equations (1) the Matlab Simulink
model was created (Figure 5). The study was set up m-file,
which also ran simulations of two INS models, with the
influence of errors and without. During the simulation
automatically were formed at the same time the differences
between two position coordinates. The coordinate’s differences
were obtained for all three axes X, Y, Z. There were performed
27 launches of Matlab Simulink model. Nine launches were
made with the assumption that no error of gyroscopes. The
accelerometers errors consistentlв taken the values 100 μg, 250
μg, 500 μg. The next nine launches INS were made with the
assumption that no error of accelerometers. The gyroscopes
errors consistentlв take the values of 1.5°/h, 2.5°/h, 5°/h. One of
the aims of this research was to determine the effect separately
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
347
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
gyroscopes and accelerometers. The final option was executed
nine studies that included the presence of errors accelerometers
and gyroscopes.
Figure 5. Matlab Simulink model of INS
In this case were formed three errors models: first model (1) 100 μg/1,5°/hr; second model (2) - 250 μg/2,5°/hr; third model
(3) - 500 μg/5°/hr. Other INS errors were left without changing
in all launches. The total time of simulation was 2 min.
Sampling frequency was 100 Hz.
5. INS SIMULATION RESULTS
Here we present the simulation results. The most comfortable is
presenting these results in graphic form. Due to lack of paper
volume, we are presenting just a little part of results. The
simpler interpretation of errors accumulation can be made for
trajectory of straight spatial line. Below presented the results of
errors accumulation along X axis.
Figure 7. Gyroscopes errors accumulation along X axis
Figure 6. Accelerometers errors accumulation along X axis
As we can see, in the modern MEMS INSs the level of errors is
too high. For distance 2 km, the displacement along X axis add
up to 1/8 from distance and grows by exponent.
Figure 8. Total errors accumulation along X axis
So, it is interesting to define in what time period the INS errors
will have accepted values. For this, we presented the graphics
of errors accumulation along Y axis for 2 seconds time interval.
As in the previous case, we are presenting displacements along
Y axis for three variants.
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
348
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
Figure 12. Changing of INS error along X axis, including wind
gust
Figure 9. Accelerometers errors accumulation along Y axis
Figure 13. Changing of INS error along Y axis, including wind
gust
Figure 10. Gyroscopes errors accumulation along Y axis
Figure 14. Changing of INS error along Z axis, including wind
gust
After this, the last step is results analysis
6. RESULTS ANALYSIS
Figure 11. Total errors accumulation along Y axis
From Figures 9-11, we can conclude that total errors of INS
have normal values. These results are very useful as allow
finding and establishing correct time interval for INS correction
by GNSS.
Another one useful property of constructed Matlab model is
possibility to research how the wind gusts can influence on
accuracy of position determination. Below presented such
simulations for model (1).
These results are quite interesting and need more dipper
analysis. Anyway, we can conclude that such atmosphere
phenomenon as wind gusts have significant influence on INS
accuracy.
To perform the analysis of the results we have to find out how
the errors of INS affecting on topographical maps accuracy.
In surveying assumed that if a source of error does not exceed
1/5 of the total value of the error, its influence is negligible. In
this case, we are considering the expected errors in scale of
aerial photographs. We can write:
X INS
where
m
X INS , YINS ,
1
5
x
Z INS
;
YINS
m
1
5
y
;
p
Z INS
H
1
5
(8)
p
= coordinates determination errors
(from INS simulation)
x
,
y
,
p
z
= errors of coordinates and parallaxes
measurements on image
m = image scale 4000 (for focus 50 mm and
surveying height 200 m)
p = parallax, 5 mm on image (for 80% overlap)
If we accept a scales of topographic maps 1:500, 1:1000,
1:2000, 1:5000, the error in determining the position will:
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
349
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
X
,
Y
0.4M ,
2
(9)
where M = map scale.
For height we will have
Z
1
h,
3
(10)
where h = vertical interval, which will take for these scales 0.5
m, 1.0 m, 1.0 m, 2.0 m.
Using the above value by expressions (9-10) we calculate the
acceptable points errors. Next, by the expressions (8) we
calculate the acceptable points errors on the image and points
errors on image due to errors INS for time interval of 2 seconds
(model 3). From the acceptable errors we move to values which
can be neglected and compare these values with INS errors.
Again, we will use simulation data for straight spatial line. The
calculation results are given in tables.
Scale
Error on map Error on image x , INS error X on
INS
,m
mm ( 0.2 x mm)
X
image, mm
500
0,14
0,035 (0,007)
0,013
1000
0,28
0,070 (0,014)
0,013
2000
0,57
0,140 (0,028)
0,013
5000
1,43
0,350 (0,070)
0,013
Table 2. Comparative analysis of calculated and acceptable INS
errors (axis X)
Scale
Error on Error on image y , INS error Y on
INS
map Y , m mm ( 0.2 mm)
image, mm
y
500
0,14
0,035 (0,007)
0,016
1000
0,28
0,070 (0,014)
0,016
2000
0,57
0,140 (0,028)
0,016
5000
1,43
0,350 (0,070)
0,016
Table 3. Comparative analysis of calculated and acceptable INS
errors (axis Y)
Error on image z , INS error
on
Error on
Z INS
Scale map
,
m
mm
(
mm)
0.2 z
Z
image, mm
500
0,17 (0,5)
0,004 (0,0008)
0,0013
1000
0,33 (1,0)
0,008 (0,0016)
0,0013
2000
0,33 (1,0)
0,008 (0,0016)
0,0013
5000
0,67 (2,0)
0,017 (0,0032)
0,0013
Table 4. Comparative analysis of calculated and acceptable INS
errors (axis Z)
If we assume that GNSS works with frequency 1 Hz and can
determine the coordinates with precision 0,02-0,03 m in any
axis, then INS allows to determine with the necessary accuracy
the position of the UAV in the intervals between the GNSSmeasurements. On the observation interval of 2 seconds INS
can be used when creating topographical maps of scale 1: 1000.
7. CONCLUSIONS
In presented paper was constructed and investigated
mathematical model of INS errors. The research established the
dominance of systematic errors of INS that accumulate during
the performance of aerial photography from UAV and require
compensation mechanism, especially for orientation angles. It
was established that low cost INS have another characteristic
feature. This is the high level of noise at the system input.
Thanks to the model developed by us, we are able to examine
separately the impact of noise in the absence of systematic
errors.
The accuracy of INS was simulated for different operating time.
For the 5 seconds time interval was established that random and
systematic impact component is almost the same. Therefore,
when performing coordinates correction by GNSS, the
assessment of systematic component impact can be done only
on time intervals from 10 seconds. At the same time the rate of
accumulation of errors in angular orientation is slower. In this
case can be recommended to use the gyroscopes errors model.
The influence of wind gusts on INS coordinates were carried
out. These results need future investigations. One of the way in
which it can be done, it is using vehicle dynamic model as it
was made in paper (Khaghani et al., 2016).
At the end of a paper was given methodic of assessment INS
errors affecting on topographical maps accuracy. Therefore we
can assess the influence of INS errors and choose proper
navigation equipment.
REFERENCES
Abdel-Hamid, W., 2005. Accuracy Enhancement of Integrated
MEMS-IMU/GPS Systems for Land Vehicular Navigation
Applications. A Thesis for the Degree Doctor of Philosophy.
Calgary.
Artese, G., Trecroci A., 2008. Calibration of a low cost MEMS
INS sensor for an integrated navigation system. In: The
International Archives of the Photogrammetry, Remote Sensing
and Spatial Information Sciences, Beijing, China, Vol.
XXXVII, Part B5, pp. 877-882.
Barret, J.M., 2014. Analyzing and modeling low-cost MEMS
IMUs for use in an inertial navigation system. A Thesis for the
Degree of Master of Science. Worcester.
Biezad, D.J., 1999. Integrated Navigation and Guidance
Systems. American Institute of Aeronautics and Astronautics,
Reston, 242 p.
Bosak, K., 2014. Secrets of UAV photomapping
http://s3.amazonaws.com/DroneMapper_US/documentation/pte
ryx-mapping-secrets.pdf.
Colomina, I., Molina, P., 2014. Unmanned aerial systems for
photogrammetry and remote sensing: A review. ISPRS Journal
of Photogrammetry and Remote Sensing, Vol. 92, pp. 79–97.
dx.doi.org/10.1015/j.isprsjprs.2014.02.013.
Goodall, C., Carmichael, S., El-Sheimy, N., Scannell, B., 2012.
INS Face Off MEMS versus FOGs. InsideGNSS,
JULY/AUGUST, pp. 48-55.
Grejner-Brzezinska, D.A., Toth, C.K., 2004. High-Accuracy
Direct Aerial Platform Orientation with Tightly Coupled
GPS/INS System. Ohio Department of Transportation, Office of
Aerial Engineering, Federal Highway Administration.
Khaghani, M., Skaloud, J., 2016. Application of vehicle
dynamic modeling in UAVs for precise determination of
exterior prientation. In: The International Archives of the
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
350
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
Photogrammetry, Remote Sensing and Spatial Information
Sciences, Prague, Czech Republic, Vol. XLI-B3, pp. 827-831.
Park, M., 2004. Error Analysis and Stochastic Modeling of
MEMS based Inertial Sensors for Land Vehicle Navigation
Applications. A Thesis for the Degree Doctor of Philosophy.
Calgary
Rehak, M., Mabillard, R., Skaloud, J., 2014. A Micro Aerial
Vehicle with Precise Position and Attitude Sensors. PFG
Photogrammetrie, Fernerkundung, Geoinformation, Issue 4, pp.
239-251, DOI: 10.1127/1432-8364/2014/0240
Salytcheva, A.O., 2004. Medium Accuracy INS/GPS
Integration in Various GPS Environments. A Thesis for the
Degree of Master of Science. Calgary.
Shin, E.-H., 2005. Estimation Techniques for Low Cost Inertial
Navigation. A Thesis for the Degree Doctor of Philosophy.
Calgary.
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Master of Science. Calgary.
Revised April 2017
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
351
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
SIMULATION OF INERTIAL NAVIGATION SYSTEM ERRORS AT AERIAL
PHOTOGRAPHY FROM UAV
R. Shults
Kyiv National University of Construction and Architecture, Faculty for Geoinformation Systems and Territory Management,
Povitroflotskyi Avenue, 31 Kyiv, 03037, Ukraine, shultsrv@gmail.com
Commission V, WG V/7
KEY WORDS: Unmanned aerial vehicle, Aerial photography, Inertial navigation system (INS), Accelerometer systematic shift,
Gyroscope systematic offset, Position accuracy
ABSTRACT:
The problem of accuracy determination of the UAV position using INS at aerial photography can be resolved in two different ways:
modelling of measurement errors or in-field calibration for INS. The paper presents the results of INS errors research by
mathematical modelling. In paper were considered the following steps: developing of INS computer model; carrying out INS
simulation; using reference data without errors, estimation of errors and their influence on maps creation accuracy by UAV data. It
must be remembered that the values of orientation angles and the coordinates of the projection centre may change abruptly due to
the influence of the atmosphere (different air density, wind, etc.). Therefore, the mathematical model of the INS was constructed
taking into account the use of different models of wind gusts. For simulation were used typical characteristics of micro
electromechanical (MEMS) INS and parameters of standard atmosphere. According to the simulation established domination of INS
systematic errors that accumulate during the execution of photographing and require compensation mechanism, especially for
orientation angles. MEMS INS have a high level of noise at the system input. Thanks to the developed model, we are able to
investigate separately the impact of noise in the absence of systematic errors. According to the research was found that on the
interval of observations in 5 seconds the impact of random and systematic component is almost the same. The developed model of
INS errors studies was implemented in Matlab software environment and without problems can be improved and enhanced with new
blocks.
1. INTRODUCTION
The last 25 years in the world, we have seen a stable increase in
requirements for geospatial data. These requirements differ but
generally this is desire to improve the accuracy and detail of
data collection and at the same time increasing the speed and
reducing cost. In the traditional version with the aim of
mapping and GIS projects data were collected using traditional
methods of terrestrial survey or aerial photography. Terrestrial
technologies are complex and energy-intensive and not very
suitable for fast and detailed data collection and updating. On
the other hand, the traditional aerial photography with long
distance from the camera to the object does not allow
completely display all its characteristics, and the results are
highly dependent on weather conditions. Both technologies are
expensive and therefore not very suitable for frequent data
updates. Today, very popular is method of data collection using
GNSS (global navigation satellite system). However, this
method is actually a continuation of traditional terrestrial
survey because it requires direct determination of each point,
although however slightly reducing the cost of work and time
spent on data collection.
An alternative to existing methods of data collection is the use
of technology, which in complex use different navigation
technologies and remote sensing (Bosak, 2014). These
technologies include aerial photography using unmanned aerial
vehicles (UAV). The main advantage of this technology is the
simultaneous reduction of cost and time for data collection. In
comparison with aerial photography, UAV camera equipment is
much simpler, the surveying distances are smaller and the
efficiency is much higher (Colomina et al. 2014). Of course,
aerial photography from the UAVs has its drawbacks, such as
low accuracy at high altitudes. However, in many projects UAV
benefits are more significant in comparison with disadvantages.
Aerial photography from the UAVs provides the accuracy that
requires the most of the topographical work. However,
achieving the required level of accuracy is challenging. The
main problems in obtaining accurate data are concern to
navigation facilities as UAV navigation unit is a complex
system with many sensors that have completely different nature
of information. UAV navigation equipment may include GNSS,
barometric sensor, magnetic compass, tilt sensor, gyroscopes
system, accelerometers systems, INS. The best solution for
aerial photography using UAVs is navigation unit GNSS/INS.
Due to the high cost and bulkiness of accurate inertial
navigation systems for UAV are used miniature
electromechanical navigation systems (MEMS). These INSs
have undeniable advantages from the point of cost and
dimensions. However, the accuracy of inertial systems wishes
to better. Accelerometers and gyroscopes that are part of the
miniature navigation systems have low measurement accuracy.
Since determination of the position and orientation in space,
using INS based on double integration of the measured
accelerations and angular velocities, even minor errors at the
beginning of integration growing very quickly over time. For
INS errors correction are using two approaches:
- The study of a particular model of INS and construction
of mathematical dependencies, which describe system errors.
- INS correction at specified intervals, using GNSS data.
The first approach is suitable for high precision, tactical-grade
INS for which errors are stable and do not change over time.
INS of this type cannot be used for UAV due to high cost and
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
huge weight. For low-cost INS rational is use of GNSS for
periodic correction (Abdel-Hamid, 2005; Grejner-Brzezinska, et
al. 2004) rather than calibration (Artese et al. 2008), due to
large and unstable errors. The choice of the interval by which
INS correction is performed depends on how quickly
accumulating errors and changing accuracy of INS in intervals
between corrections. Finally, from this depends the accuracy
with which the position of the UAV and the position and
orientation of aerial equipment will be determined (Rehak et al.,
2014). So, important is the task of INS errors simulation and
based on such research, setting INS correction interval using
GNSS. Such researches will allow choosing INS with proper
characteristics. Therefore, the main idea of this paper is
practical research of low-cost INS accuracy by mathematical
simulation results. Based on these results we tried to establish
the influence of INS position on accuracy of topographic maps
creation.
First of all we have to establish the mathematical model of INS.
c cθ
R bn s cθ
-sθ
vE
NH
vN
n
Ωen
M H
vE tg B
N H
where
ie
(1)
= Earth rotation velocity
ab = measured accelerations vector in b-system
ω bib = measured angular velocities vector in b-system
2Ω
n
ie
Ω
n
en
v
n
= vector corrections for influence of
Сoriolis and centrifugal accelerations
γ n = vector of acceleration of normal gravity
0
Ωiee 0 ; Ωien R en Ωiee
e
1
0
M H
1
D 0
N
H c B
0
0
cB
0 ,
ie s B
,
n
enz
n
enx
(3)
n
en y
where B, L, H = geodetic coordinates
vE, vN = east and north components of velocity
M, N = main radius of ellipsoid
Total rotation from e-system to n-system is describing by next
matrix:
s B c L
Ω s L
c B c L
s Bs L
cL
c Bs L
n
en
cB
0
s B
(4)
In expressions, superscript indicates the coordinate system in
which parameters are presented, and the bottom shows the
original system of coordinates. For angular velocities, lower
index indicates in respect of which two coordinate systems
occurs rotation.
Computational model of INS, which is implemented in the nsystem shown in Figure 1. Calculations in this scheme do with
equations (1).
Inertial navigation system
Angular velocities Linear accelerations
D1 v n
rn
n n b
n
n
n
n ,
v
R
a
Ω
Ω
v
γ
2
b
ie
en
n
Rb
n
b
b
R
ω
Ω
b
ib
in
(2)
Rotation from n-system to e-system can be implemented
through the transition velocity:
2. INS MODEL
The main coordinate systems, which are using in inertial
navigation, are inertial system (i-system), earth centered system
(e-system), local horizontal navigation system (n-system), body
coordinate system (b-system). The core description and
determination of these systems you can find in (Biezad , 1999;
Salytcheva, 2004).
The orientation of UAV can be determined by three angles ( φ - heading), which need for connection
roll, θ - pitch,
between vectors in b-system and n-system at the same point in
space.
We chose the INS model in which the angles and coordinates
get in n-system. In such case, we introduce a short description
of this model according to (Zang, 2003).
The equations of position and orientation calculation in nsystem have the next form:
-s cφ+c sθsφ s sφ+c sθcφ
,
c cφ+s sθsφ -c sφ+s sθcφ
cθsφ
cθcφ
a n + vn
аb
R en +
ω bib ω bnb
+
-
ω
R
-
γn
р а ь vа n
ра ітація
n
Ω en
n n
2Ωien v n Ωen
v
e
n
rn
n
Ren , Ωen
+
b
in
R bn
Ωinn
+
Ωien
R en
Ωiee
Figure 1. INS mechanisation (Zang, 2005)
Now we defined INS mathematical model. The next step is to
choose correct model for describing INS errors.
3. INS ERRORS MODEL
ie
The main measuring devices of INS are gyroscopes and
accelerometers. These devices have special errors, which
depend on structure. That is why for follow simulation is
important to choose correct INS errors model. For such model,
we have to choose measurement model for both gyroscopes and
accelerometers.
The equation of measured accelerations in matrix form will
(Park, 2004):
(5)
ab a a s a a N a a γ γ ε a ,
0
.
0
1
Rotation from b-system to n-system is describing by next
matrix:
where a = vector of clear accelerations
a = vector of accelerometers systematic bias
s a = matrix of scale coefficients
Na = matrix of angular accelerometers misalignments
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
346
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
γ = vector acceleration of normal gravity
γ = vector of acceleration of normal gravity variations
ε a = vector of accelerations measurements noise
objects) and complicated curve, which has a form of “eight”
(often is using for INS calibration). All these trajectories
presented below.
Also, equation (5) in expanded form will:
a bX a X a X s X
b
aY aY aY
aZb aZ aZ θ
γ sin θ γ X
γ cos θ sin φ γ
Y
γ cos θ cos φ γ Z
aX
aY
aZ
sY
φ
θ a X
φ aY
sZ aZ
.
The equation of measured angular velocities in matrix form
will:
(6)
ωbib ω ω s ω N ω ε ,
Figure 2. Projection of spatial line on planes XOY and XOZ
where ω = vector of clear angular velocities
ω = vector of gyroscopes systematic bias
s = matrix of scale coefficients
N = matrix of angular gyroscopes misalignments
ε = vector of gyroscopes measurements noise
Also, equation (6) in expanded form will:
b
X
b
Y
b
Z
Y
Z
X
sX
Y
θ
Z
X
sY
φ
θ
φ
sZ
Y
Z
X
Y .
Z
X
In the most cases, the random part is presented as a sum of
white noise and additional components.
(7)
W c r q t,
where
W
c
r
q
t
= white noise
= correlation noise
= random shift
= quantization noise
Figure 3. Closed route along linear object
= trembling noise
The total errors model of random and systematic errors of
accelerometers and gyroscopes for simulation are possible to
present by equations (5) and (6) using devices specifications.
4. INITIAL DATA AND MATLAB SIMULINK MODEL
FOR SIMULATION
For our simulation, we need two types of initial data: technical
specifications of INS and simulated clear accelerations and
angular velocities. In order to construct errors model we chose
follow errors values (Goodall et al. 2012; Barret, 2014), which
presented in table below.
Error
Accelerometer bias a
Axis non-orthogonality
Gyroscope bias
Accelerometers noise, RMS
Gyroscopes noise, RMS
Table 1. INS errors
Value
100 μg, 250 μg, 500 μg
60"
1,5°/hr, 2,5°/hr 5°/hr
5 mg/ hr/√Hг
0.5°/hr/√Hг
According to large value of biases, the scale coefficients of
accelerometers and gyroscopes were neglected.
In the case of measured accelerations and angular velocities
were chosen three types of trajectories: straight spatial line,
classical surveying trajectory for one route (is using for linear
Figure 4. Complicated curve
The movement of UAV has taken uniform, with a constant
speed of 80 km / h. Using equations (1) the Matlab Simulink
model was created (Figure 5). The study was set up m-file,
which also ran simulations of two INS models, with the
influence of errors and without. During the simulation
automatically were formed at the same time the differences
between two position coordinates. The coordinate’s differences
were obtained for all three axes X, Y, Z. There were performed
27 launches of Matlab Simulink model. Nine launches were
made with the assumption that no error of gyroscopes. The
accelerometers errors consistentlв taken the values 100 μg, 250
μg, 500 μg. The next nine launches INS were made with the
assumption that no error of accelerometers. The gyroscopes
errors consistentlв take the values of 1.5°/h, 2.5°/h, 5°/h. One of
the aims of this research was to determine the effect separately
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
347
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
gyroscopes and accelerometers. The final option was executed
nine studies that included the presence of errors accelerometers
and gyroscopes.
Figure 5. Matlab Simulink model of INS
In this case were formed three errors models: first model (1) 100 μg/1,5°/hr; second model (2) - 250 μg/2,5°/hr; third model
(3) - 500 μg/5°/hr. Other INS errors were left without changing
in all launches. The total time of simulation was 2 min.
Sampling frequency was 100 Hz.
5. INS SIMULATION RESULTS
Here we present the simulation results. The most comfortable is
presenting these results in graphic form. Due to lack of paper
volume, we are presenting just a little part of results. The
simpler interpretation of errors accumulation can be made for
trajectory of straight spatial line. Below presented the results of
errors accumulation along X axis.
Figure 7. Gyroscopes errors accumulation along X axis
Figure 6. Accelerometers errors accumulation along X axis
As we can see, in the modern MEMS INSs the level of errors is
too high. For distance 2 km, the displacement along X axis add
up to 1/8 from distance and grows by exponent.
Figure 8. Total errors accumulation along X axis
So, it is interesting to define in what time period the INS errors
will have accepted values. For this, we presented the graphics
of errors accumulation along Y axis for 2 seconds time interval.
As in the previous case, we are presenting displacements along
Y axis for three variants.
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
348
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
Figure 12. Changing of INS error along X axis, including wind
gust
Figure 9. Accelerometers errors accumulation along Y axis
Figure 13. Changing of INS error along Y axis, including wind
gust
Figure 10. Gyroscopes errors accumulation along Y axis
Figure 14. Changing of INS error along Z axis, including wind
gust
After this, the last step is results analysis
6. RESULTS ANALYSIS
Figure 11. Total errors accumulation along Y axis
From Figures 9-11, we can conclude that total errors of INS
have normal values. These results are very useful as allow
finding and establishing correct time interval for INS correction
by GNSS.
Another one useful property of constructed Matlab model is
possibility to research how the wind gusts can influence on
accuracy of position determination. Below presented such
simulations for model (1).
These results are quite interesting and need more dipper
analysis. Anyway, we can conclude that such atmosphere
phenomenon as wind gusts have significant influence on INS
accuracy.
To perform the analysis of the results we have to find out how
the errors of INS affecting on topographical maps accuracy.
In surveying assumed that if a source of error does not exceed
1/5 of the total value of the error, its influence is negligible. In
this case, we are considering the expected errors in scale of
aerial photographs. We can write:
X INS
where
m
X INS , YINS ,
1
5
x
Z INS
;
YINS
m
1
5
y
;
p
Z INS
H
1
5
(8)
p
= coordinates determination errors
(from INS simulation)
x
,
y
,
p
z
= errors of coordinates and parallaxes
measurements on image
m = image scale 4000 (for focus 50 mm and
surveying height 200 m)
p = parallax, 5 mm on image (for 80% overlap)
If we accept a scales of topographic maps 1:500, 1:1000,
1:2000, 1:5000, the error in determining the position will:
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
349
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
X
,
Y
0.4M ,
2
(9)
where M = map scale.
For height we will have
Z
1
h,
3
(10)
where h = vertical interval, which will take for these scales 0.5
m, 1.0 m, 1.0 m, 2.0 m.
Using the above value by expressions (9-10) we calculate the
acceptable points errors. Next, by the expressions (8) we
calculate the acceptable points errors on the image and points
errors on image due to errors INS for time interval of 2 seconds
(model 3). From the acceptable errors we move to values which
can be neglected and compare these values with INS errors.
Again, we will use simulation data for straight spatial line. The
calculation results are given in tables.
Scale
Error on map Error on image x , INS error X on
INS
,m
mm ( 0.2 x mm)
X
image, mm
500
0,14
0,035 (0,007)
0,013
1000
0,28
0,070 (0,014)
0,013
2000
0,57
0,140 (0,028)
0,013
5000
1,43
0,350 (0,070)
0,013
Table 2. Comparative analysis of calculated and acceptable INS
errors (axis X)
Scale
Error on Error on image y , INS error Y on
INS
map Y , m mm ( 0.2 mm)
image, mm
y
500
0,14
0,035 (0,007)
0,016
1000
0,28
0,070 (0,014)
0,016
2000
0,57
0,140 (0,028)
0,016
5000
1,43
0,350 (0,070)
0,016
Table 3. Comparative analysis of calculated and acceptable INS
errors (axis Y)
Error on image z , INS error
on
Error on
Z INS
Scale map
,
m
mm
(
mm)
0.2 z
Z
image, mm
500
0,17 (0,5)
0,004 (0,0008)
0,0013
1000
0,33 (1,0)
0,008 (0,0016)
0,0013
2000
0,33 (1,0)
0,008 (0,0016)
0,0013
5000
0,67 (2,0)
0,017 (0,0032)
0,0013
Table 4. Comparative analysis of calculated and acceptable INS
errors (axis Z)
If we assume that GNSS works with frequency 1 Hz and can
determine the coordinates with precision 0,02-0,03 m in any
axis, then INS allows to determine with the necessary accuracy
the position of the UAV in the intervals between the GNSSmeasurements. On the observation interval of 2 seconds INS
can be used when creating topographical maps of scale 1: 1000.
7. CONCLUSIONS
In presented paper was constructed and investigated
mathematical model of INS errors. The research established the
dominance of systematic errors of INS that accumulate during
the performance of aerial photography from UAV and require
compensation mechanism, especially for orientation angles. It
was established that low cost INS have another characteristic
feature. This is the high level of noise at the system input.
Thanks to the model developed by us, we are able to examine
separately the impact of noise in the absence of systematic
errors.
The accuracy of INS was simulated for different operating time.
For the 5 seconds time interval was established that random and
systematic impact component is almost the same. Therefore,
when performing coordinates correction by GNSS, the
assessment of systematic component impact can be done only
on time intervals from 10 seconds. At the same time the rate of
accumulation of errors in angular orientation is slower. In this
case can be recommended to use the gyroscopes errors model.
The influence of wind gusts on INS coordinates were carried
out. These results need future investigations. One of the way in
which it can be done, it is using vehicle dynamic model as it
was made in paper (Khaghani et al., 2016).
At the end of a paper was given methodic of assessment INS
errors affecting on topographical maps accuracy. Therefore we
can assess the influence of INS errors and choose proper
navigation equipment.
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This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
350
The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Volume XLII-1/W1, 2017
ISPRS Hannover Workshop: HRIGI 17 – CMRT 17 – ISA 17 – EuroCOW 17, 6–9 June 2017, Hannover, Germany
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Revised April 2017
This contribution has been peer-reviewed.
doi:10.5194/isprs-archives-XLII-1-W1-345-2017
351